Universal Quantification

Tags

logic

Is a quantification \[ \forall x \; P(x) \] which [see page 16, reads] as "for all \( x \) the predicate P(x) holds". In this expression the free-variable \( x \) is bound by the quantifier \( \forall x \).

If the predicate is true for all values then the truth-set must be the universe of discourse: \( \forall x \; P(x) \iff \mathcal{U} \). If the predicate is true for no value then it must be the empty set: \( \forall x \; \neg P(x) \iff \varnothing \).

This quantification can be [see page 20, bounded] by another set or predicate. For example \( \forall x \in A \; P(x) \) is equivalent ot \( \forall x (x \in A \implies P(x)) \).